French-American mathematician (1924?2010)
Benoit B.
[n 1]
Mandelbrot
[n 2]
(20 November 1924 ? 14 October 2010) was a Polish-born French-American
mathematician
and
polymath
with broad interests in the practical sciences, especially regarding what he labeled as "the art of
roughness
" of physical phenomena and "the uncontrolled element in life".
[6]
[7]
[8]
He referred to himself as a "fractalist"
[9]
and is recognized for his contribution to the field of
fractal geometry
, which included coining the word "fractal", as well as developing a theory of "roughness and
self-similarity
" in nature.
[10]
In 1936, at the age of 11, Mandelbrot and his family emigrated from
Warsaw
, Poland, to France. After
World War II
ended, Mandelbrot studied mathematics, graduating from universities in Paris and in the United States and receiving a master's degree in
aeronautics
from the
California Institute of Technology
. He spent most of his career in both the United States and France, having
dual
French
and
American
citizenship. In 1958, he began a 35-year career at
IBM
, where he became an
IBM Fellow
, and periodically took leaves of absence to teach at
Harvard University
. At Harvard, following the publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences.
Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovery of the
Mandelbrot set
in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order".
[11]
His math- and geometry-centered research included contributions to such fields as
statistical physics
,
meteorology
,
hydrology
,
geomorphology
,
anatomy
,
taxonomy
,
neurology
,
linguistics
,
information technology
,
computer graphics
,
economics
,
geology
,
medicine
,
physical cosmology
,
engineering
,
chaos theory
,
econophysics
,
metallurgy
, and the
social sciences
.
[12]
Toward the end of his career, he was
Sterling Professor
of Mathematical Sciences at
Yale University
, where he was the oldest professor in Yale's history to receive tenure.
[13]
Mandelbrot also held positions at the
Pacific Northwest National Laboratory
,
Universite Lille Nord de France
,
Institute for Advanced Study
and
Centre National de la Recherche Scientifique
. During his career, he received over 15 honorary doctorates and served on many science journals, along with winning numerous awards. His autobiography,
The Fractalist: Memoir of a Scientific Maverick
, was published posthumously in 2012.
Early years
[
edit
]
Benedykt Mandelbrot
[15]
was born in a
Lithuanian Jewish
family, in
Warsaw
during the
Second Polish Republic
.
[16]
His father made his living trading clothing; his mother was a dental surgeon. During his first two school years, he was tutored privately by an uncle who despised
rote learning
: "Most of my time was spent playing chess, reading maps and learning how to open my eyes to everything around me."
[17]
In 1936, when he was 11, the family emigrated from Poland to France. The move,
World War II
, and the influence of his father's brother, the mathematician
Szolem Mandelbrojt
(who had moved to Paris around 1920), further prevented a standard education. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes.
[9]
: 17
[18]
Mandelbrot attended the Lycee Rollin (now the
College-lycee Jacques-Decour
) in Paris until the start of
World War II
, when his family moved to
Tulle
, France. He was helped by
Rabbi
David Feuerwerker
, the Rabbi of
Brive-la-Gaillarde
, to continue his studies.
[9]
: 62?63
[19]
Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period:
Our constant fear was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths. This happened to a close friend from Paris,
Zina Morhange
, a physician in a nearby county seat. Simply to eliminate the competition, another physician denounced her ... We escaped this fate. Who knows why?
[9]
: 49
In 1944, Mandelbrot returned to Paris, studied at the
Lycee du Parc
in
Lyon
, and in 1945 to 1947 attended the
Ecole Polytechnique
, where he studied under
Gaston Julia
and
Paul Levy
. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics.
[2]
Returning to France, he obtained his
PhD degree
in Mathematical Sciences at the
University of Paris
in 1952.
[17]
Research career
[
edit
]
From 1949 to 1958, Mandelbrot was a staff member at the
Centre National de la Recherche Scientifique
. During this time he spent a year at the
Institute for Advanced Study
in
Princeton, New Jersey
, where he was sponsored by
John von Neumann
. In 1955 he married Aliette Kagan and moved to
Geneva, Switzerland
(to collaborate with
Jean Piaget
at the International Centre for Genetic Epistemology) and later to the
Universite Lille Nord de France
.
[20]
In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the
IBM
Thomas J. Watson Research Center
in
Yorktown Heights, New York
.
[20]
He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow
Emeritus
.
[17]
From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as
information theory
, economics, and
fluid dynamics
.
Randomness and fractals in financial markets
[
edit
]
Mandelbrot saw
financial markets
as an example of "wild randomness", characterized by concentration and long-range dependence. He developed several original approaches for modelling financial fluctuations.
[21]
In his early work, he found that the price changes in
financial markets
did not follow a
Gaussian distribution
, but rather
Levy
stable distributions
having infinite
variance
. He found, for example, that cotton prices followed a Levy stable distribution with parameter
α
equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger
scale parameter
.
[22]
The latter work from the early 60s was done with daily data of cotton prices from 1900, long before he introduced the word 'fractal'. In later years, after the concept of fractals had matured, the study of financial markets in the context of fractals became possible only after the availability of high frequency data in finance. In the late 1980s, Mandelbrot used intra-daily tick data supplied by Olsen & Associates in Zurich
[23]
[24]
to apply fractal theory to market microstructure. This cooperation led to the publication of the first comprehensive papers on scaling law in finance.
[25]
[26]
This law shows similar properties at different time scales, confirming Mandelbrot's insight of the fractal nature of market microstructure. Mandelbrot's own research in this area is presented in his books
Fractals and Scaling in Finance
[27]
and
The (Mis)behavior of Markets
.
[28]
Developing "fractal geometry" and the Mandelbrot set
[
edit
]
As a visiting professor at
Harvard University
, Mandelbrot began to study mathematical objects called
Julia sets
that were
invariant
under certain transformations of the
complex plane
. Building on previous work by
Gaston Julia
and
Pierre Fatou
, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the
Mandelbrot set
which was introduced by him in 1979.
Mandelbrot speaking about the
Mandelbrot set
, during his acceptance speech for the
Legion d'honneur
in 2006
In 1975, Mandelbrot coined the term
fractal
to describe these structures and first published his ideas in the French book
Les Objets Fractals: Forme, Hasard et Dimension
, later translated in 1977 as
Fractals: Form, Chance and Dimension
.
[29]
According to computer scientist and physicist
Stephen Wolfram
, the book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics ... to areas that had barely seen the light of serious mathematics before".
[11]
Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals":
Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea?put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.
[11]
Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole.
Fern leaves
and
Romanesque broccoli
are two examples from nature."
[11]
He points out an unexpected conclusion:
One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years?almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot.
[11]
Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphics computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960s
psychedelic art
with forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientist
Johannes Kepler
, who calculated and described the orbits of the planets.
[30]
A Mandelbrot set
Mandelbrot, however, never felt he was inventing a new idea. He described his feelings in a documentary with science writer Arthur C. Clarke:
Exploring this set I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science.
[31]
According to Clarke, "the
Mandelbrot set
is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally
infinite
complexity?" Clarke also notes an "odd coincidence":
the name Mandelbrot, and the word "
mandala
"?for a religious symbol?which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas.
[31]
In 1982, Mandelbrot expanded and updated his ideas in
The Fractal Geometry of Nature
.
[32]
This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "
program artifacts
".
Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division.
[33]
He joined the Department of Mathematics at
Yale
, and obtained his first
tenured
post in 1999, at the age of 75.
[34]
At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences.
Fractals and the "theory of roughness"
[
edit
]
Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains,
coastlines
and
river basins
; the structures of plants,
blood vessels
and
lungs
; the clustering of
galaxies
. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.
[9]
: xi
He began by asking himself various kinds of questions related to nature:
Can
geometry
deliver what the Greek root of its name [geo-] seemed to promise?truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?
[9]
: xii
In his paper "
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
", published in
Science
in 1967, Mandelbrot discusses
self-similar
curves that have
Hausdorff dimension
that are examples of
fractals
, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.
[35]
[36]
Mandelbrot emphasized the use of fractals as realistic and useful models for describing many "rough" phenomena in the real world. He concluded that "real roughness is often fractal and can be measured."
[9]
: 296
Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented in
The Fractal Geometry of Nature
had been previously described by other mathematicians. Before Mandelbrot, however, they were regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to explaining non-smooth, "rough" objects in the real world. His methods of research were both old and new:
The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer ... bringing an element of unity to the worlds of knowing and feeling ... and, unwittingly, as a bonus, for the purpose of creating beauty.
[9]
: 292
Fractals are also found in human pursuits, such as music, painting, architecture, and
stock market
prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional
Euclidean geometry
:
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
?Mandelbrot, in his introduction to
The Fractal Geometry of Nature
Section of a Mandelbrot set
Mandelbrot has been called an artist, and a visionary
[37]
and a maverick.
[38]
His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made
The Fractal Geometry of Nature
accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.
Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of
Olbers' paradox
(the "dark night sky" riddle), demonstrating the consequences of fractal theory as a
sufficient, but not necessary
, resolution of the paradox. He postulated that if the
stars
in the universe were fractally distributed (for example, like
Cantor dust
), it would not be necessary to rely on the
Big Bang
theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.
[39]
Awards and honors
[
edit
]
Mandelbrot's awards include the
Wolf Prize
for Physics in 1993, the
Lewis Fry Richardson
Prize of the
European Geophysical Society
in 2000, the
Japan Prize
in 2003,
[40]
and the Einstein Lectureship of the
American Mathematical Society
in 2006.
The small asteroid
27500 Mandelbrot
was named in his honor. In November 1990, he was made a Chevalier in France's
Legion of Honour
. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the
Pacific Northwest National Laboratory
.
[41]
Mandelbrot was promoted to an Officer of the Legion of Honour in January 2006.
[42]
An honorary degree from
Johns Hopkins University
was bestowed on Mandelbrot in the May 2010 commencement exercises.
[43]
A partial list of awards received by Mandelbrot:
[44]
Death and legacy
[
edit
]
Mandelbrot died from
pancreatic cancer
at the age of 85 in a
hospice
in
Cambridge, Massachusetts
, on 14 October 2010.
[1]
[50]
Reacting to news of his death, mathematician
Heinz-Otto Peitgen
said: "[I]f we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last fifty years."
[1]
Chris Anderson
,
TED conference
curator, described Mandelbrot as "an icon who changed how we see the world".
[51]
Nicolas Sarkozy
,
President of France
at the time of Mandelbrot's death, said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions [... h]is work, developed entirely outside mainstream research, led to modern information theory."
[52]
Mandelbrot's obituary in
The Economist
points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry".
[53]
Best-selling essayist-author
Nassim Nicholas Taleb
has remarked that Mandelbrot's book
The (Mis)Behavior of Markets
is in his opinion "The deepest and most realistic finance book ever published".
[10]
Bibliography
[
edit
]
In English
[
edit
]
- Fractals: Form, Chance and Dimension
, 1977, 2020
- The Fractal Geometry of Nature
, 1982
- Mandelbrot, Benoit B. (1983).
The Fractal Geometry of Nature
. San Francisco: W.H. Freeman.
ISBN
978-0-7167-1186-5
.
- Mandelbrot, B. (1959) Variables et processus stochastiques de Pareto-Levy, et la repartition des revenus. Comptes rendus de l'Academie des Sciences de Paris, 249, 613?615.
- Mandelbrot, B. (1960) The Pareto-Levy law and the distribution of income. International Economic Review, 1, 79?106.
- Mandelbrot, B. (1961) Stable Paretian random functions and the multiplicative variation of income. Econometrica, 29, 517?543.
- Mandelbrot, B. (1964) Random walks, fire damage amount and other Paretian risk phenomena. Operations Research, 12, 582?585.
- Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E
, 1997 by Benoit B. Mandelbrot and R.E. Gomory
- Mandelbrot, Benoit B. (1997)
Fractals and Scaling in Finance: Discontinuity, Concentration, Risk
, Springer.
- Fractales, hasard et finance
, 1959?1997, 1 November 1998
- Multifractals and 1/? Noise: Wild Self-Affinity in Physics (1963?1976)
(Selecta; V.N) 18 January 1999 by J.M. Berger and Benoit B. Mandelbrot
- Mandelbrot, Benoit (February 1999). "A Multifractal Walk down Wall Street".
Scientific American
.
280
(2): 70.
Bibcode
:
1999SciAm.280b..70M
.
doi
:
10.1038/scientificamerican0299-70
.
- Gaussian Self-Affinity and Fractals: Globality, The Earth, 1/f Noise, and R/S (Selected Works of Benoit B. Mandelbrot)
14 December 2001 by Benoit Mandelbrot and F.J. Damerau
- Mandelbrot, Benoit B.,
Gaussian Self-Affinity and Fractals
, Springer: 2002.
- Fractals and Chaos: The Mandelbrot Set and Beyond
, 9 January 2004
- The Misbehavior of Markets: A Fractal View of Financial Turbulence
, 2006 by Benoit Mandelbrot and Richard L. Hudson
- Mandelbrot, Benoit B. (2010).
The Fractalist, Memoir of a Scientific Maverick.
New York:
Vintage Books
, Division of Random House.
ISBN
978-0-307-38991-6
- The Fractalist: Memoir of a Scientific Maverick
, 2014
- Hudson, Richard L.; Mandelbrot, Benoit B. (2004).
The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward
. New York: Basic Books.
ISBN
978-0-465-04355-2
.
- Heinz-Otto Peitgen
,
Hartmut Jurgens
,
Dietmar Saupe
and Cornelia Zahlten:
Fractals: An Animated Discussion
(63 min video film, interviews with Benoit Mandelbrot and Edward Lorenz, computer animations), W.H. Freeman and Company, 1990.
ISBN
0-7167-2213-5
(re-published by Films for the Humanities & Sciences,
ISBN
978-0-7365-0520-8
)
- Mandelbrot, Benoit; Taleb, Nassim (23 March 2006).
"A focus on the exceptions that prove the rule"
.
Financial Times
. Archived from
the original
on 23 October 2010
. Retrieved
17 October
2010
.
- "Hunting the Hidden Dimension: mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature"
,
NOVA
, WGBH Educational Foundation, Boston for PBS, first aired 28 October 2008.
See also
[
edit
]
Notes
[
edit
]
References
[
edit
]
- ^
a
b
c
Hoffman, Jascha (16 October 2010).
"Benoit Mandelbrot, Mathematician, Dies at 85"
.
The New York Times
.
Archived
from the original on 18 October 2010
. Retrieved
16 October
2010
.
- ^
a
b
Lesmoir-Gordon, Nigel (17 October 2010).
"Benoit Mandelbrot obituary"
.
The Guardian
. London.
Archived
from the original on 17 September 2013
. Retrieved
17 October
2010
.
- ^
"Mandelbrot"
.
Oxford English Dictionary
(Online ed.).
Oxford University Press
.
(Subscription or
participating institution membership
required.)
- ^
Wells, John C.
(2008).
Longman Pronunciation Dictionary
(3rd ed.). Longman.
ISBN
978-1-4058-8118-0
.
- ^
Recording of the ceremony on 11 September 2006 at which Mandelbrot received the insignia for an Officer of the
Legion d'honneur
.
- ^
"Remembering the Father of Fractals"
. 22 October 2010.
Archived
from the original on 8 January 2018
. Retrieved
8 January
2018
.
- ^
Mandelbrot, Benoit (February 2010),
"Fractals and the art of roughness"
,
TED.com
, archived from
the original
on 14 April 2016
- ^
Hudson & Mandelbrot
, Prelude, page xviii
- ^
a
b
c
d
e
f
g
h
Mandelbrot, Benoit (2012).
The Fractalist: Memoir of a Scientific Maverick
. Pantheon Books.
ISBN
978-0-307-38991-6
.
- ^
a
b
Gomory, R.
(2010).
"Benoit Mandelbrot (1924?2010)"
.
Nature
.
468
(7322): 378.
Bibcode
:
2010Natur.468..378G
.
doi
:
10.1038/468378a
.
PMID
21085164
.
S2CID
4393964
.
- ^
a
b
c
d
e
Wolfram, Stephen (22 November 2012).
"The Father of Fractals"
.
The Wall Street Journal
. Archived from
the original
on 25 August 2017.
- ^
list includes specific sciences mentioned in
Hudson & Mandelbrot
, the Prelude, p. xvi, and p. 26
- ^
Olson, Steve
(November?December 2004).
"The Genius of the Unpredictable"
.
Yale Alumni Magazine
.
Archived
from the original on 22 October 2014
. Retrieved
22 July
2014
.
- ^
Mandelbrot, Benoit; Bernard Sapoval; Daniel Zajdenweber (May 1998).
"Web of Stories ? Benoit Mandelbrot ? Family background and early education"
.
Web of Stories
.
Archived
from the original on 11 September 2011
. Retrieved
19 October
2010
.
- ^
Goł?b-Meyer, Zofia (Spring 2011).
"Benoit Mandelbrot (1924?2010) ? ojciec geometrii fraktalnej"
.
Foton
.
112
. Instytut Fizyki Uniwersytetu Jagiello?skiego: 50
. Retrieved
25 December
2021
.
- ^
Hoffman, Jascha (16 October 2010).
"Benoit Mandelbrot, Novel Mathematician, Dies at 85 (Published 2010)"
.
The New York Times
.
ISSN
0362-4331
.
Archived
from the original on 21 January 2017
. Retrieved
20 November
2020
.
- ^
a
b
c
Mandelbrot, Benoit (2002).
"The Wolf Prizes for Physics,
A Maverick's Apprenticeship
"
(PDF)
. Imperial College Press.
Archived
(PDF)
from the original on 3 December 2013
. Retrieved
23 April
2012
.
- ^
"
'Fractal' mathematician Benoit Mandelbrot dies aged 85"
.
BBC Online
. 17 October 2010.
Archived
from the original on 18 October 2010
. Retrieved
17 October
2010
.
- ^
Hemenway, P. (2005).
Divine proportion: Phi in art, nature and science
. Psychology Press.
ISBN
0-415-34495-6
.
- ^
a
b
Mandelbrot, B. B. (1984).
"Mathematical People,
Interview of B. B. Mandelbrot
"
(PDF)
. Interviewed by Anthony Barcellos. Birkhauser.
Archived
(PDF)
from the original on 27 April 2015
. Retrieved
25 June
2013
.
- ^
Cont, Rama (15 May 2010). "Mandelbrot, Benoit".
Encyclopedia of Quantitative Finance
. pp. eqf01006.
doi
:
10.1002/9780470061602.eqf01006
.
ISBN
9780470057568
.
- ^
"New Scientist
, 19 April 1997"
. Newscientist.com. 19 April 1997.
Archived
from the original on 21 April 2010
. Retrieved
17 October
2010
.
- ^
Davidson, Clive (15 December 1997).
"Wildly Random Market Moves"
.
Journal of Commerce
. Archived from
the original
on 11 July 2021 – via JOC.com.
- ^
Muldoon, Oliver (14 October 2019).
"The Wandering Scientist Turned Father of Fractals"
.
Medium.com
. Retrieved
19 March
2021
.
- ^
Muller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Oliver V.; Schwarz, Matthias; Morgenegg, Claude (December 1990).
"Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis"
.
Journal of Banking and Finance
.
14
(6): 1189?1208.
doi
:
10.1016/0378-4266(90)90009-Q
– via Elsevier Science Direct.
- ^
Muller, U. A.; Dacorogna, M. M.; Dave, R. D.; Pictet, O. V.; Olsen, R. B.; Ward, J. R. (28 June 1995). "FRACTALS AND INTRINSIC TIME ? A CHALLENGE TO ECONOMETRICIANS".
Opening Lecture of the XXXIXth International Conference of the Applied Econometrics Association (AEA)
.
CiteSeerX
10.1.1.197.2969
.
- ^
Mandelbrot, Benoit (1997).
Fractals and Scaling in Finance
. Springer.
ISBN
978-1-4757-2763-0
.
- ^
Mandelbrot, Benoit (2004).
The (Mis)behavior of Markets
. Profile Books.
ISBN
9781861977656
.
- ^
Fractals: Form, Chance and Dimension
, by Benoit Mandelbrot; W H Freeman and Co, 1977;
ISBN
0-7167-0473-0
- ^
Ivry, Benjamin (17 November 2012).
"Benoit Mandelbrot Influenced Art and Mathematics"
.
The Jewish Daily Forward
. Archived from
the original
on 2 June 2013.
- ^
a
b
Arthur C Clarke ? Fractals ? The Colors Of Infinity
, archived from
the original
on 31 May 2017 – via YouTube
- ^
Mandelbrot, Benoit (1982).
The Fractal Geometry of Nature
. W H Freeman & Co.
ISBN
0-7167-1186-9
. Archived from
the original
on 30 November 2015.
- ^
Mandelbrot, Benoit; Bernard Sapoval; Daniel Zajdenweber (May 1998).
"Benoit Mandelbrot ? IBM: background and policies"
.
Web of Stories
.
Archived
from the original on 8 September 2011
. Retrieved
17 October
2010
.
- ^
Tenner, Edward (16 October 2010).
"Benoit Mandelbrot the Maverick, 1924?2010"
.
The Atlantic
.
Archived
from the original on 18 October 2010
. Retrieved
16 October
2010
.
- ^
"Benoit Mandelbrot, Novel Mathematician, Dies at 85"
.
The New York Times
. 17 October 2010. Archived from
the original
on 31 December 2018.
Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain?"
- ^
Mandelbrot, Benoit B. (5 May 1967).
"How long is the coast of Britain? Statistical self-similarity and fractional dimension"
(PDF)
.
Science
.
156
(3775): 636?638.
Bibcode
:
1967Sci...156..636M
.
doi
:
10.1126/science.156.3775.636
.
PMID
17837158
.
S2CID
15662830
.
Archived
from the original on 13 July 2015
. Retrieved
11 January
2016
.
- ^
Devaney, Robert L.
(2004).
"Mandelbrot's Vision for Mathematics"
(PDF)
.
Proceedings of Symposia in Pure Mathematics
.
72
(1). American Mathematical Society. Archived from
the original
(PDF)
on 9 December 2006
. Retrieved
5 January
2007
.
- ^
Jersey, Bill (24 April 2005).
"A Radical Mind"
.
Hunting the Hidden Dimension, NOVA
. PBS.
Archived
from the original on 22 August 2009
. Retrieved
20 August
2009
.
- ^
Gefter, Amanda (25 June 2008). "Galaxy Map Hints at Fractal Universe".
New Scientist
.
- ^
Laureates of the Japan Prize
Archived
17 April 2016 at the
Wayback Machine
. japanprize.jp
- ^
"Mandelbrot joins Pacific Northwest National Laboratory"
.
pnl.gov
(Press release). Pacific Northwest National Laboratory. 16 February 2006.
Archived
from the original on 12 January 2009
. Retrieved
17 October
2010
.
- ^
"
Legion d'honneur
announcement of promotion of Mandelbrot to
officier
"
(in French). Legifrance.gouv.fr.
Archived
from the original on 20 November 2020
. Retrieved
17 October
2010
.
- ^
"Six granted honorary degrees, Society of Scholars inductees recognized"
.
gazette.jhu.edu
. Johns Hopkins University. 7 June 2010.
Archived
from the original on 17 June 2010
. Retrieved
17 October
2010
.
- ^
Mandelbrot, Benoit B. (2 February 2006).
"Vita and Awards"
. Archived from
the original
(Word document)
on 2 July 2007
. Retrieved
15 December
2013
.
- ^
"View/Search Fellows of the ASA"
.
amstat.org
. Archived from
the original
on 16 June 2016
. Retrieved
20 August
2016
.
- ^
"APS Fellow Archive"
. APS.
Archived
from the original on 20 November 2020
. Retrieved
24 September
2020
.
- ^
"Gruppe 1: Matematiske fag"
(in Norwegian).
Norwegian Academy of Science and Letters
.
Archived
from the original on 10 November 2013
. Retrieved
7 October
2010
.
- ^
"American Philosophical Society Member History"
. Retrieved
14 June
2021
.
- ^
"Medal i Wykład im. Wacława Sierpi?skiego | Polskie Towarzystwo Matematyczne"
. Retrieved
21 January
2023
.
- ^
"Benoit Mandelbrot, fractals pioneer, dies"
.
United Press International
. 16 October 2010.
Archived
from the original on 22 October 2012
. Retrieved
17 October
2010
.
- ^
"Mandelbrot, father of fractal geometry, dies"
.
The Gazette
. Archived from
the original
on 19 October 2010
. Retrieved
16 October
2010
.
- ^
"Sarkozy rend hommage a Mandelbrot"
[Sarkozy pays homage to Mandelbrot].
Le Figaro
(in French).
Archived
from the original on 28 July 2013
. Retrieved
17 October
2010
.
- ^
Benoit Mandelbrot's obituary
Archived
24 October 2010 at the
Wayback Machine
.
The Economist
(21 October 2010)
Sources
[
edit
]
- Frame, Michael; Cohen, Nathan (2015).
Benoit Mandelbrot: A Life in Many Dimensions
. Singapore: World Scientific Publishing Company.
ISBN
978-981-4366-06-9
.
External links
[
edit
]
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